MATLAB Examples

# Arc-length control method (Crisfield, 1981 and Fafard & Massicotte, 1993)

## Notation and references

The notation followed here and in the following MATLAB codes:

• arc_length_Crisfield.m
• arc_length_Crisfield_modified.m

conforms to that used by Fafard & Massicotte in the following reference:

Fafard, M. and Massicotte, B. (1993). ”Geometrical Interpretation of the Arc-Length Method.” Computers & Structures, 46(4), 603–615. This reference is denoted as [3] inside the text of the above codes.

Except for the above study, the following reference should be noted as well:

Crisfield, M. A. (1981). ”A Fast Incremental/Iterative Solution Procedure that Handles "Snap-Through".” Computers & Structures, 13(), 55–62. This reference is denoted as [4] inside the text of the above codes.

## Algorithms implemented

1. Arc length control method as described by Fafard & Massicotte (1993), after Crisfield (1981).
2. Modified version of the above method which directs the search towards , where is the load factor.
help arc_length_Crisfield %1 help arc_length_Crisfield_modified %2 
 Arc-length control method (Crisfield, 1981) Description The equation functn(#t#)=0 is solved for #t#, where #t#=[#u#;#lambda#], #u# is the unknown displacement vector and #lambda# is the unknown load factor. The method used is the arc-length method described by Crisfield (1981): "A Fast Incremental/Iterative Solution Procedure That Handles “Snap-Through”" with the following modifications: 1.The capability to select between the cylindrical Crisfield method (original) or the spherical Crisfield method (described in [3] in the first paragraph after equation (35). 2.The initial value of #lambda# is set equal to #lambda0#+#Deltalambdabar# instead of #Deltalambdabar# as is shown in Figure 8 in [3]. The method is implemented according to the flow chart in Fig.8 and the procedure from equation (22) to equation (40) presented in [3]. Required input arguments #functn# is the function handle defining the equation to be solved. The definition of #functn# must be of the type [#R#,#Q#,#K#]=functn(#t#) where #R# ([#dim# x 1]) is the out of balance force vector, #Q# ([#dim# x 1]) is the tangent load vector given by Q(a,lambda)=-d{R(a,lambda)}/d{lambda}, #K# ([#dim# x #dim#]) is the tangent stiffness matrix given by K(a,lambda)=d{R(a,lambda)}/d{a} and #t# ([#dim#+1 x 1]) is the generalized unknown vector defined in the description section. #u0# ([#dim# x 1]) is the starting point of the solution. Optional input arguments #Crisver# (string) determines the version of the Crisfield method that will be used. It can take the values 'sph' (default) for the spherical Crisfield method or 'cyl' for the cylindrical Crisfield method (as published in [4]). #nmax# (scalar) is the maximum number of increments. Default value is 30. #Deltalambdabar# (scalar) is the load increment at the first step. Default value is 1. #imax# (scalar) is the maximum number of iterations per increment. Default value is 12. #Id# (scalar) is the desired number of iterations per increment. Default value is 4. #tol# (scalar) is the tolerance for the convergence criterion. Default value is 5e-5. #KTup# (scalar) is the stiffness matrix updater (number of iterations after which the tangent stiffness matrix is updated). For #KTup# = 1 the algorithm implemented is Full Arc-Length method. For #KTup# = Inf the algorithm implemented is Initial Stiffness Arc-Length method. Default value is 1. #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#). Default value is 1e-4. Output arguments #u# ([#dim# x #nmax#]) are the unknown displacements. #lambda# ([1 x #nmax#]) are the load factors (one per increment). #iter# ([1 x #nmax#]) is the number of iterations for each increment. #Aout# ([1 x #nmax#]) is the initial estimates of A (sign determinant) at each increment. The sign of A is positive along loading branches of the response curve (#lambda# increases) and is negative along unloading portions of the curve (#lambda# decreases). #DeltaSout# ([1 x #nmax#]) are the arc-length increments. Parents (calling functions) None. Children (called functions) None. __________________________________________________________________________ Copyright (c) 09-Mar-2014 George Papazafeiropoulos First Lieutenant, Infrastructure Engineer, Hellenic Air Force Civil Engineer, M.Sc., Ph.D. candidate, NTUA Email: gpapazafeiropoulos@yahoo.gr Website: http://users.ntua.gr/gpapazaf/ Modified arc-length control method (Crisfield, 1981) Description The equation functn(#t#)=0 is solved for #t#, where #t#=[#u#;#lambda#], #u# is the unknown displacement vector and #lambda# is the unknown load factor. The method used is the arc-length method described by Crisfield (1981): "A Fast Incremental/Iterative Solution Procedure That Handles “Snap-Through”" with the following modifications: 1.The capability to select between the cylindrical Crisfield method (original) or the spherical Crisfield method (described in [3] in the first paragraph after equation (35). 2.The initial value of #lambda# is set equal to #lambda0#+#Deltalambdabar# instead of #Deltalambdabar# as is shown in Figure (8) in [3]. 3.The solution procedure is directed towards #lambda#=1, where #lambda# is the load factor. The method is implemented according to the flow chart in Fig.8 and the procedure from equation (22) to equation (40) presented in [3]. Required input arguments #functn# is the function handle defining the equation to be solved. The definition of #functn# must be of the type [#R#,#Q#,#K#]=functn(#t#) where #R# ([#dim# x 1]) is the out of balance force vector, #Q# ([#dim# x 1]) is the tangent load vector given by Q(a,lambda)=-d{R(a,lambda)}/d{lambda}, #K# ([#dim# x #dim#]) is the tangent stiffness matrix given by K(a,lambda)=d{R(a,lambda)}/d{a} and #t# ([#dim#+1 x 1]) is the generalized unknown vector defined in the description section. #u0# ([#dim# x 1]) is the starting point of the solution. Optional input arguments #Crisver# (string) determines the version of the Crisfield method that will be used. It can take the values 'sph' (default) for the spherical Crisfield method or 'cyl' for the cylindrical Crisfield method (as published in [4]). #nmax# (scalar) is the maximum number of increments. Default value is 30. #Deltalambdabar# (scalar) is the load increment at the first step. Default value is 1. #imax# (scalar) is the maximum number of iterations per increment. Default value is 12. #Id# (scalar) is the desired number of iterations per increment. Default value is 4. #tol# (scalar) is the tolerance for the convergence criterion. Default value is 5e-5. #KTup# (scalar) is the stiffness matrix updater (number of iterations after which the tangent stiffness matrix is updated). For #KTup# = 1 the algorithm implemented is Full Arc-Length method. For #KTup# = Inf the algorithm implemented is Initial Stiffness Arc-Length method. Default value is 1. #dettol# (scalar) is the tolerance for singularity of Jacobian (#J#). Default value is 1e-4. Output arguments #u# ([#dim# x #nmax#]) are the unknown displacements. #lambda# ([1 x #nmax#]) are the load factors (one per increment). #iter# ([1 x #nmax#]) is the number of iterations for each increment. #Aout# ([1 x #nmax#]) is the initial estimates of A (sign determinant) at each increment. The sign of A is positive along loading branches of the response curve (#lambda# increases) and is negative along unloading portions of the curve (#lambda# decreases). #DeltaSout# ([1 x #nmax#]) are the arc-length increments. Parents (calling functions) None. Children (called functions) None. __________________________________________________________________________ Copyright (c) 09-Mar-2014 George Papazafeiropoulos First Lieutenant, Infrastructure Engineer, Hellenic Air Force Civil Engineer, M.Sc., Ph.D. candidate, NTUA Email: gpapazafeiropoulos@yahoo.gr Website: http://users.ntua.gr/gpapazaf/ 

Crisfield's arc-length method is described also in the material distributed to students of the Analysis and Design of Earthquake Resistant Structures postgraduate course of the School of Civil Engineering, NTUA, at the subject "Nonlinear Finite Elements" with Prof. M. Papadrakakis as course instructor. This version includes some improvements compared to the actual course material.

## Equations solved

The following equations are solved for and

## Function definitions

Two functions are utilized for the arc-length procedure:

The first function (, defined in the file function2.m ), needed to solve equation (1) is a cubic polynomial with the following properties:

• Function value:

• Function jacobian (derivative):

• Passes through the origin:

The second function (, defined in the file function1.m ), needed to solve equation (2) is a nonlinear smooth function with the following properties:

• Function value:

• Function jacobian:

## Function coding

• For function :
function [R,Q,K]=function2(t)
a=t(1:end-1);
lambda=t(end);
f1=a^3-57/8*a^2+51/4*a;
Rint=f1;
Rext=lambda*5;
% Out of balance force column vector (1-by-1)
R=Rint-Rext;
% Tangent force column vector (1-by-1)
Q=5;
% Jacobian matrix (1-by-1)
K=3*a^2-57/4*a+51/4;
end

• For function :
function [R,Q,K]=function1(t)
a=t(1:end-1);
lambda=t(end);
f1=a(1)^2+a(2)^2-49;
f2=a(1)*a(2)-24;
Rint=[f1;f2];
Rext=lambda*[1;1];
% Out of balance force column vector (2-by-1)
R=Rint-Rext;
% Tangent force column vector (2-by-1)
Q=[1;1];
% Jacobian matrix (2-by-2)
K=[2*a(1), 2*a(2);
a(2), a(1)];
end


## Initial definitions

In the subsequent code the following initial definitions are made (in the order presented below):

1. Define function
2. Define function
3. Set starting point () for solution of equation (1)
4. Set starting point () for solution of equation (2)
5. Define the version of the Crisfield method to be used (cylindrical is selected)
6. Set number of increments desired
7. Set initial value of
8. Set initial value of
9. Set maximum number of iterations permitted per increment
10. Set number of iterations desired to each converged point ()
11. Set tolerance for convergence.
12. Set the number of iterations every which the stiffness matrix of the problem is updated. KTup=1 corresponds to the full arc length method
13. Set the tolerance for determining if the stiffness matrix is singular (this is true if its determinant is below dettol)
functn1=@function2; %1 functn2=@function1; %2 u01=0.1; %3 u02=[4;6]; %4 Crisver='cyl'; %5 nmax=30; %6 lambda0=0; %7 Deltalambdabar=0.4; %8 imax=20; %9 Id=1; %10 tol=5e-5; %11 KTup=1; %12 dettol=1e-4; %13 

## Applications

1. Default application of the arc length control method as described by Crisfield (1981) to solve equation (1)
2. Default application of the modified version of the Crisfield (1981) arc length control method to solve equation (1)
3. Non-default application of the arc length control method as described by Crisfield (1981) to solve equation (1)
4. Non-default application of the modified version of the Crisfield (1981) arc length control method to solve equation (1)
5. Default application of the arc length control method as described by Crisfield (1981) to solve equation (2)
6. Default application of the modified version of the Crisfield (1981) arc length control method to solve equation (2)
7. Non-default application of the arc length control method as described by Crisfield (1981) to solve equation (2)
8. Non-default application of the modified version of the Crisfield (1981) arc length control method to solve equation (2)
[u1,lambda1,iter1,Aout1,DeltaSout1] = arc_length_Crisfield(functn1,u01); %1 Result1=[u1',lambda1',iter1',Aout1',DeltaSout1'] %1 [u2,lambda2,iter2,Aout2,DeltaSout2] = arc_length_Crisfield_modified(functn1,u01); %2 Result2=[u2',lambda2',iter2',Aout2',DeltaSout2'] %2 [u3,lambda3,iter3,Aout3,DeltaSout3] = arc_length_Crisfield(functn1,u01,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %3 Result3=[u3',lambda3',iter3',Aout3',DeltaSout3'] %3 [u4,lambda4,iter4,Aout4,DeltaSout4] = arc_length_Crisfield_modified(functn1,u01,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %4 Result4=[u4',lambda4',iter4',Aout4',DeltaSout4'] %4 [u5,lambda5,iter5,Aout5,DeltaSout5] = arc_length_Crisfield(functn2,u02); %5 Result5=[u5',lambda5',iter5',Aout5',DeltaSout5'] %5 [u6,lambda6,iter6,Aout6,DeltaSout6] = arc_length_Crisfield_modified(functn2,u02); %6 Result6=[u6',lambda6',iter6',Aout6',DeltaSout6'] %6 [u7,lambda7,iter7,Aout7,DeltaSout7] = arc_length_Crisfield(functn2,u02,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %7 Result7=[u7',lambda7',iter7',Aout7',DeltaSout7'] %7 [u8,lambda8,iter8,Aout8,DeltaSout8] = arc_length_Crisfield_modified(functn2,u02,Crisver,nmax,lambda0,Deltalambdabar,imax,Id,tol,KTup,dettol); %8 Result8=[u8',lambda8',iter8',Aout8',DeltaSout8'] %8 
Result1 = 1.0e+12 * 0.0000 0.0000 0.0000 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0000 0.0000 0.0003 0.0000 0.0000 0.0000 0.0000 0.0007 0.0000 0.0000 0.0000 0.0000 0.0017 0.0000 0.0000 0.0000 0.0000 0.0042 0.0000 0.0000 0.0000 0.0000 0.0099 0.0000 0.0000 0.0000 0.0000 0.0234 0.0000 0.0000 0.0000 0.0000 0.0556 0.0000 0.0000 0.0000 0.0000 0.1318 0.0000 0.0000 0.0000 0.0000 0.3128 0.0000 0.0000 0.0000 0.0000 0.7418 0.0000 0.0000 0.0000 0.0000 1.7589 0.0000 0.0000 0.0000 0.0000 4.1706 0.0000 0.0000 0.0000 Result2 = -0.9927 1.3440 12.0000 0 1.0927 -0.6284 -2.2149 3.0000 -0.1830 0.3642 -0.1428 -0.3938 3.0000 0.0796 0.4856 0.5047 3.7003 12.0000 0.1636 0.6475 0.7205 1.3895 12.0000 0.5121 0.2158 0.6486 1.1090 3.0000 0.2671 0.0719 0.5526 1.0078 3.0000 -0.0754 0.0959 0.6806 1.2117 12.0000 -0.0828 0.1279 0.6379 1.0987 3.0000 0.1440 0.0426 0.5811 1.0398 3.0000 -0.0437 0.0568 0.6569 1.1419 12.0000 -0.0518 0.0758 0.6316 1.0925 3.0000 0.0809 0.0253 0.5979 1.0580 3.0000 -0.0255 0.0337 0.5530 1.0082 3.0000 -0.0318 0.0449 0.6129 1.0895 12.0000 -0.0388 0.0599 0.5929 1.0527 3.0000 0.0582 0.0200 0.5663 1.0234 3.0000 -0.0186 0.0266 0.6018 1.0676 12.0000 -0.0236 0.0355 0.5900 1.0495 3.0000 0.0337 0.0118 0.5742 1.0322 2.0000 -0.0110 0.0158 0.6057 1.0706 12.0000 -0.0142 0.0315 0.5952 1.0551 3.0000 0.0302 0.0105 0.5812 1.0400 2.0000 -0.0099 0.0140 0.5532 1.0084 3.0000 -0.0128 0.0280 0.5906 1.0563 12.0000 -0.0242 0.0374 0.5781 1.0365 3.0000 0.0347 0.0125 0.5615 1.0179 2.0000 -0.0113 0.0166 0.5947 1.0594 12.0000 -0.0146 0.0332 0.5836 1.0426 3.0000 0.0311 0.0111 0.5689 1.0263 2.0000 -0.0102 0.0148 Result3 = 0.2761 0.5997 2.0000 0 0.1761 0.3642 0.7494 2.0000 0.0974 0.0881 0.4082 0.8171 2.0000 0.0553 0.0440 0.4303 0.8493 2.0000 0.0296 0.0220 0.4413 0.8649 2.0000 0.0153 0.0110 0.4468 0.8727 1.0000 0.0078 0.0055 0.4523 0.8803 1.0000 0.0039 0.0055 0.4578 0.8879 1.0000 0.0040 0.0055 0.4633 0.8954 1.0000 0.0040 0.0055 0.4688 0.9028 1.0000 0.0041 0.0055 0.4743 0.9102 1.0000 0.0041 0.0055 0.4798 0.9175 1.0000 0.0041 0.0055 0.4853 0.9248 1.0000 0.0042 0.0055 0.4908 0.9319 1.0000 0.0042 0.0055 0.4963 0.9390 1.0000 0.0042 0.0055 0.5018 0.9460 1.0000 0.0043 0.0055 0.5073 0.9530 1.0000 0.0043 0.0055 0.5128 0.9599 1.0000 0.0044 0.0055 0.5183 0.9667 1.0000 0.0044 0.0055 0.5238 0.9735 1.0000 0.0045 0.0055 0.5293 0.9802 1.0000 0.0045 0.0055 0.5348 0.9868 1.0000 0.0046 0.0055 0.5403 0.9934 1.0000 0.0046 0.0055 0.5458 0.9998 1.0000 0.0046 0.0055 0.5513 1.0063 1.0000 0.0047 0.0055 0.5568 1.0126 1.0000 0.0047 0.0055 0.5624 1.0189 1.0000 0.0048 0.0055 0.5679 1.0251 1.0000 0.0048 0.0055 0.5734 1.0313 1.0000 0.0049 0.0055 0.5789 1.0374 1.0000 0.0049 0.0055 Result4 = 0.2761 0.5997 2.0000 0 0.1761 0.3642 0.7494 2.0000 0.0974 0.0881 0.4082 0.8171 2.0000 0.0553 0.0440 0.4303 0.8493 2.0000 0.0296 0.0220 0.4413 0.8649 2.0000 0.0153 0.0110 0.4468 0.8727 1.0000 0.0078 0.0055 0.4523 0.8803 1.0000 0.0039 0.0055 0.4578 0.8879 1.0000 0.0040 0.0055 0.4633 0.8954 1.0000 0.0040 0.0055 0.4688 0.9028 1.0000 0.0041 0.0055 0.4743 0.9102 1.0000 0.0041 0.0055 0.4798 0.9175 1.0000 0.0041 0.0055 0.4853 0.9248 1.0000 0.0042 0.0055 0.4908 0.9319 1.0000 0.0042 0.0055 0.4963 0.9390 1.0000 0.0042 0.0055 0.5018 0.9460 1.0000 0.0043 0.0055 0.5073 0.9530 1.0000 0.0043 0.0055 0.5128 0.9599 1.0000 0.0044 0.0055 0.5183 0.9667 1.0000 0.0044 0.0055 0.5238 0.9735 1.0000 0.0045 0.0055 0.5293 0.9802 1.0000 0.0045 0.0055 0.5348 0.9868 1.0000 0.0046 0.0055 0.5403 0.9934 1.0000 0.0046 0.0055 0.5458 0.9998 1.0000 0.0046 0.0055 0.5403 0.9935 20.0000 0.0047 0.0055 0.5406 0.9937 2.0000 -0.0046 0.0003 0.5407 0.9938 1.0000 0.0002 0.0001 0.5409 0.9940 1.0000 0.0001 0.0001 0.5410 0.9942 1.0000 0.0001 0.0001 0.5412 0.9943 1.0000 0.0001 0.0001 Result5 = 4.6983 5.2551 0.6900 4.0000 0 1.0210 5.3835 4.4982 0.2161 4.0000 0.6445 1.0210 5.7208 3.5345 -3.7801 4.0000 -0.4127 1.0210 5.7581 2.5141 -9.5235 4.0000 -0.1995 1.0210 5.5806 1.5086 -15.5810 4.0000 -0.1706 1.0210 5.2493 0.5428 -21.1505 4.0000 -0.1760 1.0210 4.8020 -0.3750 -25.8006 4.0000 -0.2017 1.0210 4.2622 -1.2417 -29.2922 3.0000 -0.2540 1.0210 3.4237 -2.3142 -31.9230 3.0000 -0.3641 1.3614 2.1272 -3.5846 -31.6253 3.0000 -1.3020 1.8152 0.1211 -4.9384 -24.5978 4.0000 1.3221 2.4202 -2.1700 -5.7184 -11.5910 4.0000 0.5553 2.4202 -4.5614 -5.3459 0.3846 5.0000 0.3984 2.4202 -5.6740 -3.7613 -2.6585 4.0000 1.0337 1.9362 -5.6562 -1.8252 -13.6764 4.0000 -0.3901 1.9362 -5.0015 -0.0031 -23.9847 5.0000 -0.3211 1.9362 -4.2041 1.3249 -29.5699 4.0000 -0.4296 1.5489 -3.2301 2.5292 -32.1696 4.0000 -0.5776 1.5489 -2.1100 3.5991 -31.5942 3.0000 -2.3411 1.5489 -0.4153 4.7794 -25.9848 4.0000 1.1052 2.0653 1.4903 5.5757 -15.6906 4.0000 0.5189 2.0653 3.5506 5.7180 -3.6975 4.0000 0.3547 2.0653 5.3041 4.6269 0.5413 5.0000 0.3995 2.0653 5.7707 3.0419 -6.4462 4.0000 -1.0001 1.6522 5.5520 1.4042 -16.2037 4.0000 -0.2831 1.6522 4.9346 -0.1283 -24.6330 4.0000 -0.2857 1.6522 4.0574 -1.5284 -30.2013 4.0000 -0.3807 1.6522 2.9861 -2.7862 -32.3200 4.0000 -0.6988 1.6522 1.7539 -3.8868 -30.8169 3.0000 -8.7454 1.6522 -0.1134 -5.0557 -23.4269 4.0000 0.8261 2.2029 Result6 = 5.0110 5.8573 6.2854 12.0000 0 1.0210 4.6371 5.2973 0.5642 10.0000 0.4007 0.3403 4.7482 5.2186 0.7788 3.0000 -0.0000 0.1361 4.8890 5.1040 0.9538 3.0000 0.1033 0.1815 4.6995 5.2545 0.6920 12.0000 0.2989 0.2420 4.7642 5.2064 0.8045 3.0000 -0.1561 0.0807 4.8482 5.1391 0.9153 3.0000 0.0651 0.1076 4.7470 5.2408 0.9182 12.0000 0.1312 0.1434 4.7723 5.2002 0.8169 3.0000 -0.1027 0.0478 4.8222 5.1606 0.8855 3.0000 0.0373 0.0637 4.8871 5.1057 0.9522 3.0000 0.0669 0.0850 4.8075 5.1863 0.9582 12.0000 0.1377 0.1133 4.8288 5.1552 0.8935 3.0000 -0.1057 0.0378 4.8674 5.1228 0.9348 3.0000 0.0395 0.0504 4.9178 5.0784 0.9742 3.0000 0.0699 0.0672 4.8550 5.1422 0.9813 12.0000 0.1480 0.0895 4.8727 5.1182 0.9397 3.0000 -0.1101 0.0298 4.9027 5.0920 0.9642 3.0000 0.0420 0.0398 4.9418 5.0562 0.9869 3.0000 0.0744 0.0531 4.8924 5.1067 0.9939 12.0000 0.1641 0.0707 4.9069 5.0882 0.9671 3.0000 -0.1166 0.0236 4.9302 5.0670 0.9813 2.0000 0.0454 0.0314 4.8856 5.1114 0.9805 12.0000 0.0813 0.0629 4.8990 5.0953 0.9615 3.0000 -0.0985 0.0210 4.9197 5.0766 0.9754 2.0000 0.0374 0.0279 4.9606 5.0385 0.9939 2.0000 0.0631 0.0559 4.8780 5.1137 0.9444 12.0000 0.2540 0.1118 4.9059 5.0891 0.9665 3.0000 -0.1680 0.0373 4.9426 5.0555 0.9872 3.0000 0.0720 0.0497 4.8961 5.1027 0.9923 12.0000 0.1556 0.0662 Result7 = 3.8946 5.6384 -2.0406 5.0000 0 0.0825 3.9105 5.6337 -1.9696 2.0000 0.0000 0.0165 3.9184 5.6314 -1.9344 1.0000 0.0038 0.0082 3.9263 5.6290 -1.8993 1.0000 0.0019 0.0082 3.9341 5.6266 -1.8643 1.0000 0.0019 0.0082 3.9420 5.6241 -1.8295 1.0000 0.0020 0.0082 3.9499 5.6217 -1.7949 1.0000 0.0020 0.0082 3.9578 5.6192 -1.7605 1.0000 0.0020 0.0082 3.9656 5.6167 -1.7262 1.0000 0.0020 0.0082 3.9735 5.6142 -1.6920 1.0000 0.0020 0.0082 3.9813 5.6117 -1.6581 2.0000 0.0020 0.0082 3.9852 5.6104 -1.6412 1.0000 0.0020 0.0041 3.9892 5.6091 -1.6243 1.0000 0.0010 0.0041 3.9931 5.6078 -1.6074 1.0000 0.0010 0.0041 3.9970 5.6065 -1.5907 1.0000 0.0010 0.0041 4.0009 5.6052 -1.5739 1.0000 0.0010 0.0041 4.0048 5.6039 -1.5572 1.0000 0.0010 0.0041 4.0087 5.6026 -1.5405 1.0000 0.0010 0.0041 4.0126 5.6013 -1.5239 1.0000 0.0010 0.0041 4.0165 5.6000 -1.5073 1.0000 0.0010 0.0041 4.0204 5.5987 -1.4908 1.0000 0.0010 0.0041 4.0244 5.5973 -1.4743 1.0000 0.0010 0.0041 4.0283 5.5960 -1.4579 1.0000 0.0010 0.0041 4.0322 5.5947 -1.4415 1.0000 0.0010 0.0041 4.0360 5.5933 -1.4251 1.0000 0.0010 0.0041 4.0399 5.5920 -1.4088 1.0000 0.0010 0.0041 4.0438 5.5906 -1.3925 1.0000 0.0010 0.0041 4.0477 5.5892 -1.3763 1.0000 0.0010 0.0041 4.0516 5.5879 -1.3601 1.0000 0.0010 0.0041 4.0555 5.5865 -1.3439 1.0000 0.0011 0.0041 Result8 = 3.8946 5.6384 -2.0406 5.0000 0 0.0825 3.9105 5.6337 -1.9696 2.0000 0.0000 0.0165 3.9184 5.6314 -1.9344 1.0000 0.0038 0.0082 3.9263 5.6290 -1.8993 1.0000 0.0019 0.0082 3.9341 5.6266 -1.8643 1.0000 0.0019 0.0082 3.9420 5.6241 -1.8295 1.0000 0.0020 0.0082 3.9499 5.6217 -1.7949 1.0000 0.0020 0.0082 3.9578 5.6192 -1.7605 1.0000 0.0020 0.0082 3.9656 5.6167 -1.7262 1.0000 0.0020 0.0082 3.9735 5.6142 -1.6920 1.0000 0.0020 0.0082 3.9813 5.6117 -1.6581 2.0000 0.0020 0.0082 3.9852 5.6104 -1.6412 1.0000 0.0020 0.0041 3.9892 5.6091 -1.6243 1.0000 0.0010 0.0041 3.9931 5.6078 -1.6074 1.0000 0.0010 0.0041 3.9970 5.6065 -1.5907 1.0000 0.0010 0.0041 4.0009 5.6052 -1.5739 1.0000 0.0010 0.0041 4.0048 5.6039 -1.5572 1.0000 0.0010 0.0041 4.0087 5.6026 -1.5405 1.0000 0.0010 0.0041 4.0126 5.6013 -1.5239 1.0000 0.0010 0.0041 4.0165 5.6000 -1.5073 1.0000 0.0010 0.0041 4.0204 5.5987 -1.4908 1.0000 0.0010 0.0041 4.0244 5.5973 -1.4743 1.0000 0.0010 0.0041 4.0283 5.5960 -1.4579 1.0000 0.0010 0.0041 4.0322 5.5947 -1.4415 1.0000 0.0010 0.0041 4.0360 5.5933 -1.4251 1.0000 0.0010 0.0041 4.0399 5.5920 -1.4088 1.0000 0.0010 0.0041 4.0438 5.5906 -1.3925 1.0000 0.0010 0.0041 4.0477 5.5892 -1.3763 1.0000 0.0010 0.0041 4.0516 5.5879 -1.3601 1.0000 0.0010 0.0041 4.0555 5.5865 -1.3439 1.0000 0.0011 0.0041